HIERARCHICAL CRITERIA 345

The table of \n(n — i) intercolumnar correlations offers much
the same problems as the initial table of \n(n — i) observed
correlations. The number of items now correlated (ri) will almost
invariably be much smaller than the number of persons correlated
for the initial table (N) : only in one or two researches does it exceed
3O.1 Hence, in testing significance, we should in strictness adopt
the principles proposed by recent statistical writers for testing small
samples. In at least two respects, however, a set of intercolumnar
correlations will differ from ordinary correlations : the sampling
distribution of the variables correlated, namely, correlation co-
efficients, is not normal; and the degrees of freedom will be still
further restricted. If we retain this line of approach, it would not
be difficult to modify the usual proofs to take both these peculiarities
into account. But, as I have indicated below, if we are aiming at
great precision, a somewhat different mode of attack would seem
desirable. However, with the rough data available in psychology,
precise determinations are out of the question. Hence, in an
elementary discussion we may ignore these further refinements.

On the other hand, with a chance distribution of the coefficients
we may reasonably assume that both the means and the standard
deviations of all the columns would be alike. In particular, with a
table of residual correlations (for which we chiefly need our criteria)
the means, as obtained by the simple summation method (the
' centroid method,' as Thurstone terms it) are zero. It is true, as we
have just seen, that the average intercolumnar correlation (even when
the correlated coefficients are not distributed at random) will also
be zero : for half the correlations will be numerically identical with
the reversed half, but will have opposite signs. But the propor-
tionality criterion will remain unaffected if, following the device
familiarized by Thurstone and used by him in testing the significance
of his residual tables, the variables for one half of the table are
reversed in sign.

When the means and standard deviations of each column
are identical, there is, as we have already seen, a very simple

1 Notably in the recent remarkable study by Thurstone to which reference
will frequently be made in the sequel [122]. Here 57 tests applied to 240
persons were correlated for the initial correlation table. With the method
of correlation employed, however, the standard error is said to be about 'Oo,
(p. 61). Thurstone does not apply his earlier suggestions to demonstrate
whether the correlation matrix is of rank one (e.g. the intercolumnar or
proportionality criteria [15], pp. 134-5), although, as we shall see, almost
all the latent roots except the first are of doubtful significance.